# Risk Analysis: Changing the Story with the Statistical Stochastic Process and VaR

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. A Statistical Risk Analysis under Uncertainty: Stochastic Process and Simulation

_{t}represents future revenues, and $\frac{d{R}_{t}}{{R}_{t}}$ is the movement of uncertain future revenues in a small amount of time t. ${\alpha}_{t}$ is the drift in the movement of revenue; ${\sigma}_{t}$ is the volatility of the future revenues; and ${z}_{1}$ is a random variable whose probability distribution is normal. Formula (3) is the famous Brownian stochastic process, in which uncertainty is described with both “expected revenues” and “possible variance of revenues”.

## 4. Conclusions and Managerial Implications

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Value at Risk (shadow area) for the actual project value. Broken line: The trends for possible dynamics. Dotted line: The normal distribution.

**Figure 2.**All possible paths of uncertain benefits of Project D. One color represents for one possible path. Figures stands for all possible paths of uncertain benefits of Project D.

Project A | Project B | Project C | Project D | Project E | |
---|---|---|---|---|---|

Expected Revenues | $5000 | $2000 | $3000 | $10,000 | $6000 |

Costs | $4000 | $3000 | $2000 | $7786 | $4000 |

Net Present Value | $1000 | −$1000 | $1000 | $2214 | $2000 |

Uncertainty Degree (Ignored in the NPV analysis) | 50% | 50% | 50% | 50% | 80% |

Priority Decision | 3 | 5 | 3 | 1 | 2 |

Project A | Project B | Project C | Project D | Project E | |
---|---|---|---|---|---|

Expected Revenues | $5000 | $2000 | $3000 | $10,000 | $6000 |

Uncertainty Degree | 50% | 50% | 50% | 50% | 80% |

Costs | $4000 | $3000 | $2000 | $7786 | $4000 |

Net Present Value | $1000 | −$1000 | $1000 | $2214 | $2000 |

Expected Revenues under Uncertainty (VaR 90%) | $3974 | $1646 | $2470 | $8256 | $4256 |

Monetary Exposed Uncertainty | −$1026 | −$354 | −$530 | −$1744 | −$1744 |

Net Present Value under Uncertainty | −$26 | −$1354 | $470 | $470 | $256 |

Return per Uncertainty Unit | −3% | −383% | 88.7% | 27% | 15% |

Decision under NPV | 3 | 5 | 3 | 1 | 2 |

Decision under VaR | 4 | 5 | 1 | 2 | 3 |

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**MDPI and ACS Style**

Wu, L.
Risk Analysis: Changing the Story with the Statistical Stochastic Process and VaR. *Axioms* **2023**, *12*, 418.
https://doi.org/10.3390/axioms12050418

**AMA Style**

Wu L.
Risk Analysis: Changing the Story with the Statistical Stochastic Process and VaR. *Axioms*. 2023; 12(5):418.
https://doi.org/10.3390/axioms12050418

**Chicago/Turabian Style**

Wu, Lianghong.
2023. "Risk Analysis: Changing the Story with the Statistical Stochastic Process and VaR" *Axioms* 12, no. 5: 418.
https://doi.org/10.3390/axioms12050418